After spending more time with MizuDroid, I think I understand why both options produce the same result. In "Settings", "Run in background" is enabled. Enabling background mode may explain why tapping the "Exit" button produces the same result as tapping the "Run in background" button.

However, it does not explain the following behavior:

In "Settings" I disabled "Run in background".

Tap Android's "Back" icon on phone.

MizuDroid displays a dialog-box asking "Are you sure you want to close MizuDroid?" and the options are "Cancel" and "Exit".

Tapping either button produces the same result: MizuDroid appears in the "Recents" list and has not exited (not shutdown and removed from memory).

I don't understand the designer's intentions for the "Exit" button. Regardless of mode or which button is tapped, MizuDroid appears to behave the same way (it never truly exits).